document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=808848>Question 808848</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the equation for exact solution over the interval [0,360) \r<BR>\n" );
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document.write( "tan &#920; csc &#920; - &#8730;3 csc &#920; = 0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #487246 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=487246\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \tan\varphi\ =\ \frac{\sin\varphi}{\cos\varphi}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \csc\varphi\ =\ \frac{1}{\sin\varphi}\">\r<BR>\n" );
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document.write( "So\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \tan\theta\csc\theta\ -\ \sqrt{3}\csc\theta\ =\ 0\">\r<BR>\n" );
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document.write( "is equivalent to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{1}{\cos\theta}\ -\ \frac{\sqrt{3}}{\sin\theta}\ =\ 0\">\r<BR>\n" );
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document.write( "Add\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{\sqrt{3}}{\sin\theta}\">\r<BR>\n" );
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document.write( "to both sides:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{1}{\cos\theta}\ =\ \frac{\sqrt{3}}{\sin\theta}\">\r<BR>\n" );
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document.write( "Cross multiply:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \sin\theta\ =\ \sqrt{3}\cos\theta\">\r<BR>\n" );
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document.write( "Square both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \sin^2\theta\ =\ 3\cos^2\theta\">\r<BR>\n" );
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document.write( "Use the Pythagorean identity\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \sin^2\theta\ =\ 3\ -\ 3\sin^2\theta\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 4\sin^2\theta\ =\ 3\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \sin\theta\ =\ \pm\frac{\sqrt{3}}{2}\">\r<BR>\n" );
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document.write( "From which we get, using the unit circle, that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \theta\ =\ \frac{\pi}{3},\,\frac{2\pi}{3},\,\frac{4\pi}{3},\,\frac{5\pi}{3}\">\r<BR>\n" );
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document.write( "Which then implies that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \cos\theta\ =\ \pm\frac{1}{2}\">\r<BR>\n" );
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document.write( "But if sine and cosine are opposite signs, the original equation fails.  Hence, the solution set is confined to Quadrants I and III where the signs of sine and cosine are the same.  And the solution set is therefore:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left{\theta\ \in\ \mathbb{R}\ |\ \theta =\ \frac{\pi}{3} \ \large\vee\LARGE\ \theta\ =\ \frac{4\pi}{3}\right}\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "<font face=\"Math1\" size=\"+2\">Egw to Beta kai to Sigma</font><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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